Change of Variables & the Jacobian

Substitution in multiple integrals via the Jacobian determinant.

The idea

Suppose $x = g(u,v)$ and $y = h(u,v)$ map a region $S$ of the $uv$-plane one-to-one onto a region $R$ of the $xy$-plane. The Jacobian determinant of that map is $\frac{\partial(x,y)}{\partial(u,v)} = \det\begin{pmatrix} x_u & x_v \\ y_u & y_v \end{pmatrix} = x_u\,y_v - x_v\,y_u.$

Theorem (Change of variables).

If $x = g(u,v)$, $y = h(u,v)$ map $S$ one-to-one onto $R$, with continuous partial derivatives and a Jacobian that vanishes at most on the boundary of $S$, then $\iint_R f(x,y)\,dA = \iint_S f\bigl(g(u,v),h(u,v)\bigr)\,\left|\frac{\partial(x,y)}{\partial(u,v)}\right|\,du\,dv.$

This is the substitution rule of one-variable calculus, where $dx$ became $g'(u)\,du$ and the derivative acted as a stretching factor. In the plane the factor measures area. A small rectangle with sides $du$ and $dv$ is generally not sent to a rectangle: moving $u$ by $du$ displaces the image point by about $\langle x_u, y_u\rangle\,du$, and moving $v$ by $dv$ displaces it by about $\langle x_v, y_v\rangle\,dv$, so the image is a small parallelogram with those two edges, of area $|x_u y_v - x_v y_u|\,du\,dv$. The Jacobian is the local area-scaling factor, and its absolute value keeps the two sides of the theorem measuring the same amount of area.

We use the theorem to trade an awkward region for a simple one: a substitution that turns a disc into a rectangle, or a slanted parallelogram into a square, usually repays a messier integrand.

Ways to work on it

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